Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

Abstract

<p style='text-indent:20px;'>We study the quasilinear elliptic equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -Qu = e^u ~~~~\mbox{in}~~~~ \Omega\subset \mathbb{R}^{N}, \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where the operator <inline-formula><tex-math id="M1">\begin{document}$ Q $\end{document}</tex-math></inline-formula>, known as the Finsler-Laplacian (or anisotropic Laplacian) operator, is defined by <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>Here <inline-formula><tex-math id="M2">\begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ F: \mathbb{R}^{N}\rightarrow[0, +\infty) $\end{document}</tex-math></inline-formula> is a convex function of <inline-formula><tex-math id="M4">\begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document}</tex-math></inline-formula> that satisfies certain assumptions. For a bounded domain <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed <inline-formula><tex-math id="M6">\begin{document}$ N-10 $\end{document}</tex-math></inline-formula>. For the case of entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions <inline-formula><tex-math id="M7">\begin{document}$ N&lt;10 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ 2&lt;N&lt;10 $\end{document}</tex-math></inline-formula>, respectively. We also provide an explicit solution that is stable outside a compact set in two dimensions <inline-formula><tex-math id="M9">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula>. In addition, we present similar Liouville theorems for the related equations with power-type nonlinearities.

Topics & Concepts